Question

a guy wire runs from the top of a cell tower to a metal stake in the ground. lincoln places a 10 - foot tall pole to support the guy wire. after placing the pole, lincoln measures the distance from the stake to the pole to be 5 ft. he then measures the distance from the pole to the tower to be 14 ft. find the length of the guy wire, to the nearest foot.

Explanation:

Step1: Set up the proportion

Since the two triangles (formed by the pole - stake and tower - stake) are similar. Let the height of the tower be \(h\). The proportion is \(\frac{h}{10}=\frac{14 + 5}{5}\).

Step2: Solve for the height of the tower

Cross - multiply: \(5h=10\times(14 + 5)\).
\(5h=10\times19\).
\(5h = 190\).
\(h=\frac{190}{5}=38\) feet.

Step3: Use the Pythagorean theorem to find the length of the guy wire

Let the length of the guy wire be \(l\). The base of the large right - triangle (with the tower) is \(14 + 5=19\) feet and the height is \(h = 38\) feet.
By the Pythagorean theorem \(l=\sqrt{38^{2}+19^{2}}\).
\(l=\sqrt{1444+361}\).
\(l=\sqrt{1805}\).
\(l\approx42.5\).

Answer:

\(43\) feet.